Continuity envelopes of spaces of generalised smoothness: a limiting case; embeddings and approximation numbers

Abstract

Continuity envelopes for the spaces of generalised smoothness Bpq(s,Ψ)(ℝn) and Fpq(s,Ψ)(ℝn) are studied in the so-called supercritical s=1+n/p, paralleling recent developments for a corresponding limiting case for local growth envelopes of spaces of such a type. In addition, the power of the concept is used in proving conditions for some embeddings between function spaces to hold, as well as in the study of the asymptotic behaviour of approximation numbers of related embeddings.